link or maybe separation problem
Internally, when you create a linked duplicate, blender does not actually duplicate the object, but it has another reference to it, thus, when you edit the objects, you are editing all linked duplicates the same way. You need to break the link  between the two if you want to edit those elements separately. 

Since I am pretty bad at explaining these sort of things without demonstrating, here are two helpful links about duplicates and linked duplicates. 

I would advise you to look at these two links How to unlink a linked-duplicate? and Duplication -- Blender manual.